Chaos in coupled lateral-longitudinal vibration of electrostatically actuated microresonators

Abstract

The precision and stability of the microresonators can be affected by their nonlinear vibrational behaviors. So, the main objective of this paper is to predict the nonlinear dynamic phenomena in the electrostatically actuated microresonators. In the framework of the Von Karman's theory and the Euler-Bernoulli beam model, the nonlinear governing equations of motion are developed using the Hamilton's principle. Also, the coupling of lateral and longitudinal vibrations is considered. The governing partial differential equations are discretized by means of the single-mode Galerkin's method and solved using the Runge–Kutta method. The influences of various system parameters such as the actuation frequency, lateral-longitudinal coupling, amplitude of the ac voltage and the flexural rigidity on the response of the microresonator are investigated. The bifurcation diagrams, power spectra analysis, Poincare’ map, phase plane portrait, maximum Lyapunov exponent and the Melnikov function are employed to inspect the chaotic behavior of the microresonator. The results indicate that, the vibrational behaviors of the microresonator include 2T-periodic, 4T-periodic, 7T-periodic, quasi-periodic and chaotic motions. Also, at low actuation frequency, considering the longitudinal coupling effects makes a substantial difference in the routes to chaos and vibrational response of microresonator. So, optimization and performance improvements of microresonator can be realized by the results of this paper.